Properties

Label 2-8015-1.1-c1-0-217
Degree $2$
Conductor $8015$
Sign $-1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.81·2-s − 0.874·3-s + 1.27·4-s − 5-s + 1.58·6-s + 7-s + 1.30·8-s − 2.23·9-s + 1.81·10-s − 4.01·11-s − 1.11·12-s + 3.22·13-s − 1.81·14-s + 0.874·15-s − 4.92·16-s + 1.62·17-s + 4.04·18-s − 4.71·19-s − 1.27·20-s − 0.874·21-s + 7.27·22-s + 0.483·23-s − 1.14·24-s + 25-s − 5.83·26-s + 4.57·27-s + 1.27·28-s + ⋯
L(s)  = 1  − 1.27·2-s − 0.504·3-s + 0.638·4-s − 0.447·5-s + 0.646·6-s + 0.377·7-s + 0.463·8-s − 0.745·9-s + 0.572·10-s − 1.21·11-s − 0.322·12-s + 0.893·13-s − 0.483·14-s + 0.225·15-s − 1.23·16-s + 0.394·17-s + 0.953·18-s − 1.08·19-s − 0.285·20-s − 0.190·21-s + 1.55·22-s + 0.100·23-s − 0.233·24-s + 0.200·25-s − 1.14·26-s + 0.881·27-s + 0.241·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8015\)    =    \(5 \cdot 7 \cdot 229\)
Sign: $-1$
Analytic conductor: \(64.0000\)
Root analytic conductor: \(8.00000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 - T \)
229 \( 1 - T \)
good2 \( 1 + 1.81T + 2T^{2} \)
3 \( 1 + 0.874T + 3T^{2} \)
11 \( 1 + 4.01T + 11T^{2} \)
13 \( 1 - 3.22T + 13T^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 + 4.71T + 19T^{2} \)
23 \( 1 - 0.483T + 23T^{2} \)
29 \( 1 + 7.46T + 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 - 8.72T + 37T^{2} \)
41 \( 1 - 2.50T + 41T^{2} \)
43 \( 1 - 3.80T + 43T^{2} \)
47 \( 1 - 6.39T + 47T^{2} \)
53 \( 1 - 4.28T + 53T^{2} \)
59 \( 1 - 5.68T + 59T^{2} \)
61 \( 1 - 0.252T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 - 6.03T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 5.95T + 83T^{2} \)
89 \( 1 - 1.33T + 89T^{2} \)
97 \( 1 + 9.36T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.57375819921264103694187908283, −7.20546653490586299021698925468, −6.00877536275057668892904466250, −5.65279797288710084896501458125, −4.67520406738963294182070701876, −3.98793270751809707814952465326, −2.84966242374402092480384585526, −1.99153963251898467452226893519, −0.859523045529503653893959527249, 0, 0.859523045529503653893959527249, 1.99153963251898467452226893519, 2.84966242374402092480384585526, 3.98793270751809707814952465326, 4.67520406738963294182070701876, 5.65279797288710084896501458125, 6.00877536275057668892904466250, 7.20546653490586299021698925468, 7.57375819921264103694187908283

Graph of the $Z$-function along the critical line